ON EMBEDDINGS AND THE H-CLOSEDNESS OF TOPOLOGICAL SEMIGROUPS OF PARTIAL FINITE BIJECTIONS

All topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [1, 2, 3]. A topological space X is called countably compact if any countable open cover of X contains a finite subcover [3]. A topological space X is called pseudocompact if each continuous real-valued function on X is bounded [3]. A (topological) semigroup is a set with a (continuous) binary associative opera-tion. An element e of a semigroup S is called an idempotent if e · e = e. The subset of idempotents of a semigroup S is called the band of S end denoted by E(S). Let λ be a cardinal> 1. On the set Bλ = λ × λ ∪ {0} we define the semigroup operation as follows (α, β) · (γ, δ) = (α, δ), if β = γ; 0, if β 6 = γ, and (α, β) · 0 = 0 · (α, β) = 0 · 0 = 0, for all α, β, γ, δ ∈ λ. The semigroup Bλ is called the semigroup of λ×λ-matrix units [2]. Let I(X) denote the set of all partial one-to-one transformations of X together with the following semigroup operation